Digital twins of road systems require high-fidelity finite element (FE) simulations that can be executed fast enough to support prediction, monitoring, and decision-making under evolving loading and environmental conditions. Achieving this performance is particularly challenging for inelastic material models, where history-dependent state variables must accurately evolve at the material-point level. Therefore, a multi-layer reduction strategy for fast FE simulations tailored to digital twins of road infrastructures is presented. First, an Arbitrary Lagrangian Eulerian (ALE) formulation is employed to confine the computational domain to a moving region of interest, significantly reducing the number of active degrees of freedom while retaining full spatial resolution in critical zones. Building on this reduced mesh, a hyper-reduced order model based on energy-conserving sampling and weighting is created to further accelerate the computations, leading to the final hyper-reduced domain that is evaluated during the FE simulation. A key challenge addressed in this work is the treatment of inelastic materials under ALE kinematics, where advection of material points leads to a non-trivial evolution of history variables. To this end, we reconstruct the material history using a gappy proper orthogonal decomposition framework, enabling accurate recovery of internal variables on the whole domain based on the material point information of the hyper-reduced domain. Numerical examples relevant to road systems demonstrate substantial speed-ups compared to full-order simulations, while maintaining accuracy in both global responses and local inelastic behavior. The proposed approach provides an efficient and robust building block for real-time-capable digital twins of road systems, bridging the gap between high-fidelity inelastic FE simulations and operational performance requirements.